refunds? The card debited immediately when had to be charged to the registration and now I have never paid.

I had made a hotel booking without a fully refundable cancellation fee and had it canceled 33 days before the deadline. They say that I have never paid and will not repay money. This was clearly loaded on my card. I wrote to and it was said that they will have to open an investigation that will take 2 billing cycles. Someone experienced something like this?